Optimal. Leaf size=57 \[ \frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}} \]
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Rubi [A] time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {96, 93, 204} \[ \frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 96
Rule 204
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x}}+\frac {3}{7} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x}}+\frac {6}{7} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x}}-\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{7 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 57, normalized size = 1.00 \[ \frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.47, size = 71, normalized size = 1.25 \[ -\frac {33 \, \sqrt {7} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 28 \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{539 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.15, size = 100, normalized size = 1.75 \[ \frac {3}{490} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {4 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{385 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 108, normalized size = 1.89 \[ \frac {\left (66 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-33 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-28 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{539 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,\left (3\,x+2\right )\,\sqrt {5\,x+3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right ) \sqrt {5 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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